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Mirrors > Home > HSE Home > Th. List > bdophsi | Structured version Visualization version GIF version |
Description: The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoptri.1 | ⊢ 𝑆 ∈ BndLinOp |
nmoptri.2 | ⊢ 𝑇 ∈ BndLinOp |
Ref | Expression |
---|---|
bdophsi | ⊢ (𝑆 +op 𝑇) ∈ BndLinOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoptri.1 | . . . 4 ⊢ 𝑆 ∈ BndLinOp | |
2 | bdopln 30845 | . . . 4 ⊢ (𝑆 ∈ BndLinOp → 𝑆 ∈ LinOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝑆 ∈ LinOp |
4 | nmoptri.2 | . . . 4 ⊢ 𝑇 ∈ BndLinOp | |
5 | bdopln 30845 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ 𝑇 ∈ LinOp |
7 | 3, 6 | lnophsi 30985 | . 2 ⊢ (𝑆 +op 𝑇) ∈ LinOp |
8 | bdopf 30846 | . . . . . 6 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 ⊢ 𝑆: ℋ⟶ ℋ |
10 | bdopf 30846 | . . . . . 6 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | |
11 | 4, 10 | ax-mp 5 | . . . . 5 ⊢ 𝑇: ℋ⟶ ℋ |
12 | 9, 11 | hoaddcli 30752 | . . . 4 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
13 | nmopxr 30850 | . . . 4 ⊢ ((𝑆 +op 𝑇): ℋ⟶ ℋ → (normop‘(𝑆 +op 𝑇)) ∈ ℝ*) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (normop‘(𝑆 +op 𝑇)) ∈ ℝ* |
15 | nmopre 30854 | . . . . 5 ⊢ (𝑆 ∈ BndLinOp → (normop‘𝑆) ∈ ℝ) | |
16 | 1, 15 | ax-mp 5 | . . . 4 ⊢ (normop‘𝑆) ∈ ℝ |
17 | nmopre 30854 | . . . . 5 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) | |
18 | 4, 17 | ax-mp 5 | . . . 4 ⊢ (normop‘𝑇) ∈ ℝ |
19 | 16, 18 | readdcli 11175 | . . 3 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ |
20 | nmopgtmnf 30852 | . . . 4 ⊢ ((𝑆 +op 𝑇): ℋ⟶ ℋ → -∞ < (normop‘(𝑆 +op 𝑇))) | |
21 | 12, 20 | ax-mp 5 | . . 3 ⊢ -∞ < (normop‘(𝑆 +op 𝑇)) |
22 | 1, 4 | nmoptrii 31078 | . . 3 ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)) |
23 | xrre 13094 | . . 3 ⊢ ((((normop‘(𝑆 +op 𝑇)) ∈ ℝ* ∧ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ) ∧ (-∞ < (normop‘(𝑆 +op 𝑇)) ∧ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)))) → (normop‘(𝑆 +op 𝑇)) ∈ ℝ) | |
24 | 14, 19, 21, 22, 23 | mp4an 692 | . 2 ⊢ (normop‘(𝑆 +op 𝑇)) ∈ ℝ |
25 | elbdop2 30855 | . 2 ⊢ ((𝑆 +op 𝑇) ∈ BndLinOp ↔ ((𝑆 +op 𝑇) ∈ LinOp ∧ (normop‘(𝑆 +op 𝑇)) ∈ ℝ)) | |
26 | 7, 24, 25 | mpbir2an 710 | 1 ⊢ (𝑆 +op 𝑇) ∈ BndLinOp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 class class class wbr 5106 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 ℝcr 11055 + caddc 11059 -∞cmnf 11192 ℝ*cxr 11193 < clt 11194 ≤ cle 11195 ℋchba 29903 +op chos 29922 normopcnop 29929 LinOpclo 29931 BndLinOpcbo 29932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-hilex 29983 ax-hfvadd 29984 ax-hvcom 29985 ax-hvass 29986 ax-hv0cl 29987 ax-hvaddid 29988 ax-hfvmul 29989 ax-hvmulid 29990 ax-hvmulass 29991 ax-hvdistr1 29992 ax-hvdistr2 29993 ax-hvmul0 29994 ax-hfi 30063 ax-his1 30066 ax-his2 30067 ax-his3 30068 ax-his4 30069 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-grpo 29477 df-gid 29478 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-nmcv 29584 df-hnorm 29952 df-hba 29953 df-hvsub 29955 df-hosum 30714 df-nmop 30823 df-lnop 30825 df-bdop 30826 |
This theorem is referenced by: bdophdi 31081 nmoptri2i 31083 unierri 31088 |
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